Single-Phase a.c. Calculations

A single-phase circuit containing a resistor, an inductor and a capacitor is shown in Figure 12.1. You will recall that the phase relationship between the voltage and the current in a circuit element depends on the nature of the element, in other words, is it an R or an L or a C? This means that in an a.c. circuit you cannot simply add the numerical values of V _R , V _L and V _C together to get the value of the supply voltage V _S ; the reason for this is that the voltage phasors representing V _R , V _L and V _C ‘point’ in different directions relative to the current on the phasor diagram. To account for the differing ‘directions’ of the phasors, you have to calculate V _S as the phasor sum of the three component voltages in Figure 12.1. That is <m:math display='block'> <m:semantics> <m:mrow> <m:math display='block'> <m:mrow> <m:mtext>supply&#x00A0;voltage,&#x00A0;</m:mtext><m:msub> <m:mi>V</m:mi> <m:mi>S</m:mi> </m:msub> <m:mtext>&#x00A0;=&#x00A0;</m:mtext><m:mi>p</m:mi><m:mi>h</m:mi><m:mi>a</m:mi><m:mi>s</m:mi><m:mi>o</m:mi><m:mi>r</m:mi><m:mi>&#x0020;</m:mi><m:mi>s</m:mi><m:mi>u</m:mi><m:mi>m</m:mi><m:mtext>&#x00A0;of&#x00A0;</m:mtext><m:msub> <m:mi>V</m:mi> <m:mi>R</m:mi> </m:msub> <m:mtext>&#x00A0;,&#x00A0;</m:mtext><m:msub> <m:mi>V</m:mi> <m:mi>L</m:mi> </m:msub> <m:mtext>&#x00A0;and&#x00A0;</m:mtext><m:msub> <m:mi>V</m:mi> <m:mi>C</m:mi> </m:msub> </m:mrow> </m:math> </m:semantics> </m:math> $$ {\text{supply voltage, }}{V_S}{\text{ = }}phasor sum{\text{ of }}{V_R}{\text{ , }}{V_L}{\text{ and }}{V_C} $$ To illustrate how this is applied to the circuit in Figure 12.1, consider the case where the current, I, is 1.5 A, and the three voltages are <m:math display='block'> <m:semantics> <m:mrow> <m:math display='block'> <m:mrow> <m:msub> <m:mi>V</m:mi> <m:mi>R</m:mi> </m:msub> <m:mtext>&#x00A0;=150&#x00A0;V,&#x00A0;</m:mtext><m:msub> <m:mi>V</m:mi> <m:mi>L</m:mi> </m:msub> <m:mtext>&#x00A0;=&#x00A0;200&#x00A0;V,&#x00A0;</m:mtext><m:msub> <m:mi>V</m:mi> <m:mi>C</m:mi> </m:msub> <m:mtext>&#x00A0;=&#x00A0;100&#x00A0;V</m:mtext></m:mrow> </m:math> </m:semantics> </m:math> $$ {V_R}{\text{ = 150 V, }}{V_L}{\text{ = 200 V, }}{V_C}{\text{ = 100 V}} $$ We shall consider in turn the phasor diagram for each element, after which we shall combine them to form- the phasor diagram for the complete circuit. fig 12.1an R-L-C series circuit